A mathematical theory of stochastic microlensing. II. Random images, shear, and the Kac-Rice formula

Continuing our development of a mathematical theory of stochastic microlensing, we study the random shear and expected number of random lensed images of different types. In particular, we characterize the first three leading terms in the asymptotic expression of the joint probability density function (pdf) of the random shear tensor due to point masses in the limit of an infinite number of stars. Up to this order, the pdf depends on the magnitude of the shear tensor, the optical depth, and the mean number of stars through a combination of radial position and the star's mass. As a consequence, the pdf's of the shear components are seen to converge, in the limit of an infinite number of stars, to shifted Cauchy distributions, which shows that the shear components have heavy tails in that limit. The asymptotic pdf of the shear magnitude in the limit of an infinite number of stars is also presented. All the results on the random microlensing shear are given for a general point in the lens plane. Extending to the general random distributions (not necessarily uniform) of the lenses, we employ the Kac-Rice formula and Morse theory to deduce general formulas for the expected total number of images and the expected number of saddle images. We further generalize these results by considering random sources defined on a countable compact covering of the light source plane. This is done to introduce the notion of global expected number of positive parity images due to a general lensing map. Applying the result to microlensing, we calculate the asymptotic global expected number of minimum images in the limit of an infinite number of stars, where the stars are uniformly distributed. This global expectation is bounded, while the global expected number of images and the global expected number of saddle images diverge as the order of the number of stars. © 2009 American Institute of Physics.

Muscle Logic: New Knowledge Resource for Anatomy Enables Comprehensive Searches of the Literature on the Feeding Muscles of Mammals.

BACKGROUND: In recent years large bibliographic databases have made much of the published literature of biology available for searches. However, the capabilities of the search engines integrated into these databases for text-based bibliographic searches are limited. To enable searches that deliver the results expected by comparative anatomists, an underlying logical structure known as an ontology is required. DEVELOPMENT AND TESTING OF THE ONTOLOGY: Here we present the Mammalian Feeding Muscle Ontology (MFMO), a multi-species ontology focused on anatomical structures that participate in feeding and other oral/pharyngeal behaviors. A unique feature of the MFMO is that a simple, computable, definition of each muscle, which includes its attachments and innervation, is true across mammals. This construction mirrors the logical foundation of comparative anatomy and permits searches using language familiar to biologists. Further, it provides a template for muscles that will be useful in extending any anatomy ontology. The MFMO is developed to support the Feeding Experiments End-User Database Project (FEED, https://feedexp.org/), a publicly-available, online repository for physiological data collected from in vivo studies of feeding (e.g., mastication, biting, swallowing) in mammals. Currently the MFMO is integrated into FEED and also into two literature-specific implementations of Textpresso, a text-mining system that facilitates powerful searches of a corpus of scientific publications. We evaluate the MFMO by asking questions that test the ability of the ontology to return appropriate answers (competency questions). We compare the results of queries of the MFMO to results from similar searches in PubMed and Google Scholar. RESULTS AND SIGNIFICANCE: Our tests demonstrate that the MFMO is competent to answer queries formed in the common language of comparative anatomy, but PubMed and Google Scholar are not. Overall, our results show that by incorporating anatomical ontologies into searches, an expanded and anatomically comprehensive set of results can be obtained. The broader scientific and publishing communities should consider taking up the challenge of semantically enabled search capabilities.